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An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function

Author

Listed:
  • Omran, A.K.
  • Zaky, M.A.
  • Hendy, A.S.
  • Pimenov, V.G.

Abstract

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear Riesz-space and Caputo-time fractional reaction–diffusion equation with prehistory. The problem is first approximated by the L1 difference method in the temporal direction, and then the Galerkin–Legendre spectral method is applied for the spatial discretization. The key advantage of the proposed method is that the implementation of the iterative approach is linear. The stability and the convergence of the semi-discrete approximation are proved by invoking the discrete fractional Halanay inequality. The stability and convergence of the fully discrete scheme are also investigated utilizing discrete fractional Grönwall inequalities, which show that the proposed method is stable and convergent. Furthermore, to verify the efficiency of our method, we provide numerical results that show a satisfactory agreement with the theoretical analysis.

Suggested Citation

  • Omran, A.K. & Zaky, M.A. & Hendy, A.S. & Pimenov, V.G., 2022. "An easy to implement linearized numerical scheme for fractional reaction–diffusion equations with a prehistorical nonlinear source function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 218-239.
    • Handle: RePEc:eee:matcom:v:200:y:2022:i:c:p:218-239
    DOI: 10.1016/j.matcom.2022.04.014
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    References listed on IDEAS

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    1. Zhang, Qifeng & Ren, Yunzhu & Lin, Xiaoman & Xu, Yinghong, 2019. "Uniform convergence of compact and BDF methods for the space fractional semilinear delay reaction–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 91-110.
    2. Hafez, Ramy M. & Zaky, Mahmoud A. & Hendy, Ahmed S., 2021. "A novel spectral Galerkin/Petrov–Galerkin algorithm for the multi-dimensional space–time fractional advection–diffusion–reaction equations with nonsmooth solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 678-690.
    3. Zhao, Yong-Liang & Zhu, Pei-Yong & Luo, Wei-Hua, 2018. "A fast second-order implicit scheme for non-linear time-space fractional diffusion equation with time delay and drift term," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 231-248.
    4. Mahmoud A. Zaky & Ahmed S. Hendy & Rob H. De Staelen, 2021. "Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System," Mathematics, MDPI, vol. 9(2), pages 1-22, January.
    5. Bellen, Alfredo & Zennaro, Marino, 2013. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780199671373.
    6. Hao, Zhaopeng & Fan, Kai & Cao, Wanrong & Sun, Zhizhong, 2016. "A finite difference scheme for semilinear space-fractional diffusion equations with time delay," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 238-254.
    7. Liu, Pan-Ping, 2015. "Periodic solutions in an epidemic model with diffusion and delay," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 275-291.
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    More about this item

    Keywords

    Fractional reaction–diffusion; Prehistory; L1 difference scheme; Galerkin–Legendre spectral method; Fractional Halanay inequalities; Discrete fractional Grönwall inequalities;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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